Number Theory

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The Riemann Hypothesis is a Consequence of CT-Invariant Quantum Mechanics

Authors:Carlos CastroComments: 17 pages, This article appeared in the Int. Jour. of Geom. Methods of Modern Physics vol 5, no. 1, February 2008

The Riemann's hypothesis (RH) states that the nontrivial zeros of the
Riemann zeta-function are of the form sn = 1/2 + iλn. By constructing
a continuous family of scaling-like operators involving the Gauss-Jacobi
theta series and by invoking a novel CT-invariant Quantum Mechanics,
involving a judicious charge conjugation C and time reversal T operation,
we show why the Riemann Hypothesis is true. An infinite family of theta
series and their Mellin transform leads to the same conclusions.
Category:Number Theory

Polynomials with Rational Roots that Differ by a Non-zero Constant

The problem of finding two polynomials P(x) and Q(x) of a given degree n in a single variable
x that have all rational roots and differ by a non-zero constant is investigated. It is shown
that the problem reduces to considering only polynomials with integer roots. The cases n < 4
are solved generically. For n = 4 the case of polynomials whose roots come in pairs (a,-a) is
solved. For n = 5 an infinite number of inequivalent solutions are found with the ansatz
P(x) = -Q(-x). For n = 6 an infinite number of solutions are also found. Finally for n = 8
we find solitary examples. This also solves the problem of finding two polynomials of degree
n that fully factorise into linear factors with integer coefficients such that the difference
is one.
Category:Number Theory

On the Riemann Hypothesis, Area Quantization, Dirac Operators, Modularity and Renormalization Group

Two methods to prove the Riemann Hypothesis are presented. One is
based on the modular properties of Θ (theta) functions and the other on
the Hilbert-Polya proposal to find an operator whose spectrum reproduces
the ordinates ρn (imaginary parts) of the zeta zeros in the critical line :
sn = 1/2 + iρn
A detailed analysis of a one-dimensional Dirac-like operator
with a potential V(x) is given that reproduces the spectrum of energy levels
En = ρn, when the boundary conditions
ΨE (x = -∞) = ± ΨE (x = +∞) are imposed.
Such potential V(x) is derived implicitly from the
relation x = x(V) = π/2(dN(V)/dV), where the functional form of N(V)
is given by the full-fledged Riemann-von Mangoldt counting function of
the zeta zeros, including the fluctuating as well as the O(E-n) terms.
The construction is also extended to self-adjoint Schroedinger operators.
Crucial is the introduction of an energy-dependent cut-off function Λ(E).
Finally, the natural quantization of the phase space areas (associated to
nonperiodic crystal-like structures) in integer multiples of π follows from
the Bohr-Sommerfeld quantization conditions of Quantum Mechanics. It
allows to find a physical reasoning why the average density of the primes
distribution for very large x (O(1/logx)) has a one-to-one correspondence
with the asymptotic limit of the inverse average density of the zeta zeros
in the critical line suggesting intriguing connections to the Renormalization
Group program.
Category:Number Theory

A Proof for Goldbach's Conjecture

For a large even number there are a large number of pairs of odd
numbers sum of the members of each being the even number. We eliminate
those pairs that none of the members of each of them is prime and show
that the number of the remaining pairs is still large. The process of proof
shows that there can be no drop to zero in the function of the number of
the mentioned prime pairs.
Category:Number Theory